Π-trivial map

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Contents

[edit] 1 Introduction

This page is based on [Ranicki2002]. A map f:N^n\to M^m between manifolds represents a homology class f_*[N] \in H_n(M). Let (\widetilde{M},\pi,w) be an oriented cover with covering map p:\widetilde{M} \to M. If f factors through \widetilde{M}/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Bx22oy as f= p\circ \widetilde{f}: N \to \widetilde{M}\to M then f represents a homology class \widetilde{f}_*[N]\in H_n(\widetilde{M}). Note that a choice of lift \widetilde{f} is required in order to represent a homology class.

Let b_1 be a basepoint of N. By covering space theory (c.f. [Hatcher2002, Proposition 1.33]) a map f:N\to M can be lifted to \widetilde{M} if and only if f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)})), i.e. if and only if the composition q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi is trivial for q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)}) the quotient map.

The group \pi is well-defined for any choice of lift \widetilde{f(b_1)} since p:\widetilde{M}\to M is a regular covering and changing the basepoint in \widetilde{M} to a different lift corresponds to conjugating \pi_1(\widetilde{M},\widetilde{f(b_1)}) by some g\in \pi_1(M,f(b_1)).

[edit] 2 Definition

Let M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_qAQ3yy be an m-dimensional manifold and let (\widetilde{M},\pi,w) be an oriented cover. A \pi-trivial map f:N^n\to M^m is a map from an oriented manifold N with basepoint b_1 such that the composite

\displaystyle \xymatrix{   \pi_1(N,b_1) \ar[r]^-{f_*} & \pi_1(M,f(b_1)) \ar[r] & \pi  }

is trivial, together with a choice of lift \widetilde{f}:N \to \widetilde{M}.

[edit] 3 Properties

A map f:N\to M that factors through \widetilde{M} must map all of N to the same sheet of \widetilde{M}, hence the pullback satisfies
\displaystyle f^*\widetilde{M} \cong N\times \pi.
Choosing where to lift a single point determines a lift \widetilde{f}:N \to \widetilde{M}, which thought of as a map from N\times \{1\} \subset N \times \pi extends equivariantly to a lift \widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}.

[edit] 4 Lifts and paths - two alternative perspectives

Rather than taking a lift as part of the data for a \pi-trivial map we could instead take an equivalence class of paths in M as is explained in this section. Since \pi is the group of deck transformations of \widetilde{M}, the set of lifts \{\widetilde{f}:N\to \widetilde{M}\} is non-canonically isomorphic to \pi with the group structure determined by the action of \pi once a choice of lift \widetilde{f}_{\!\id} has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a \pi-trivial map can be thought of as a choice of isomorphism

\displaystyle  \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g \; \; \mathrm{s.t.}\; \widetilde{f}(b_1) = g\widetilde{f}_{\!\id}(b_1).\end{array}
Let b be a basepoint of M. The set of homotopy classes of paths from b to f(b_1) is non-canonically isomophic to \pi_1(M,b). An isomorphism is defined by a choice of path [w_{id}] to represent the identity element:
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where * denotes concatenation of paths and w_{id}^{-1} is the path w_{id} in reverse. Let \widetilde{b} be a basepoint of \widetilde{M} that is a lift of b. Define an equivalence relation \sim on this set by saying
\displaystyle [w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).
The above isomorphism given by choosing [w_{id}] descends to give an isomorphism
\displaystyle \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,
where we use the same choice of path [w_{id}] to identify \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b}) with \pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi. Thus a choice of lift \widetilde{f}:N\to \widetilde{M} corresponds to a choice of homotopy class of paths from b to f(b_1) modulo \pi_1(\widetilde{M}). A choice of lift \widetilde{b} defines a bijection of sets
\displaystyle \{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim
as follows. Given a choice of lift \widetilde{f} choose any path \widetilde{w}:I \to \widetilde{M} from \widetilde{b} to \widetilde{f}(b_1). Take the equivalence class of p(\widetilde{w}) which is a path in M from b to f(b_1). Conversely given a choice of class
\displaystyle [w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim
choose any representative w:I \to M. This lifts uniquely to a path \widetilde{w} starting at \widetilde{b}. Define a lift \widetilde{f} by setting \widetilde{f}(b_1):= \widetilde{w}(1). Note this map is well-defined since different choices of representative w may differ by elements of \pi_1(\widetilde{M},\widetilde{b}) but their lifts will still end at the same point.

To sum up we have the following diagram of non-canonical isomorphisms and bijections

\displaystyle \xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \\ & \pi & }.
Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups.

Since an oriented cover comes with a choice of lift \widetilde{b} as part of the data a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a \pi-trivial map.

[edit] 5 Examples

Let f:S^n\looparrowright M^{2n} be an immersion and let \widetilde{M} be the universal cover of M, let s\in S^n and b\in M be basepoints. For n>1, \pi_1(S^n)=0 so f lifts to \widetilde{M}. An immersion f:S^n\looparrowright M^{2n} is a \pi_1(M)-trivial immersion as soon as a lift \widetilde{f}:S^n \looparrowright \widetilde{M} has been prescribed or, alternatively, once a homotopy class of paths w:I\to M from b to f(s) has been prescribed. A pair (f:S^n\looparrowright M^{2n}, w: I\to M | w(0)=b \; \text{and} \; w(1)=f(s)) is often called a pointed immersion in the literature (See, for example, [Lück2001, Section 4.1]).

[edit] 6 References

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